VOLUME 89, N UMBER 23
PHYSICA L R EVIEW LET T ERS
2 DECEMBER 2002
Control of Light Pulse Propagation with Only a Few Cold Atoms in a High-Finesse Microcavity
Yukiko Shimizu,* Noritsugu Shiokawa,† Noriaki Yamamoto, Mikio Kozuma,‡ and Takahiro Kuga
Institute of Physics, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan
L. Deng and E.W. Hagley
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
(Received 5 June 2002; published 15 November 2002)
Propagation of a light pulse through a high-Q optical microcavity containing a few cold atoms
(N < 10) in its cavity mode is investigated experimentally. With less than ten cold rubidium atoms
launched into an optical microcavity, up to 170 ns propagation lead time (‘‘superluminal’’), and 440 ns
propagation delay time (subluminal) are observed. Comparison of the experimental data with numerical
simulations as well as future experiments are discussed.
DOI: 10.1103/PhysRevLett.89.233001
When a well-defined electromagnetic disturbance
propagates in a highly dispersive medium, its group
velocity can be described by vg c=n !dn=d!,
where c is the speed of the electromagnetic wave in
vacuum, and ! is the wave’s angular frequency. This
expression of the group velocity indicates that the functional form of the refractive index n and its derivative
determine the group velocity at which the wave propagates. In the frequency region where the index changes
rapidly [1–8], the group velocity can become very different from the speed of light in vacuum [9], and sometimes
may even result in apparent ‘‘superluminal’’ propagation
[10]. The studies reported thus far on superluminal and
subluminal propagation of optical pulses have been all
carried out with highly dense atomic systems such as
atomic vapor cells at elevated temperatures, and recently
with Bose-Einstein condensed alkali vapors. A key element in these studies is that a significantly large number
of atoms (> 106 ) interact with the probe pulse. Because
the refractive index is intrinsically a macroscopic quantity, the statistical ensemble average will lead to the direct
proportionality between the refractive index and the total
number of the atoms with which the probe pulse interacts
as it propagates through the medium. To our knowledge,
there has been no study of abnormal wave propagation
effects using a high-finesse cavity to enhance the dispersive properties of an individual atom.
When a resonant or near resonant optical field (or even
a single photon) propagates through a cavity containing a
single atom, the atomic and probe light states become
highly entangled. Such an entangled state, also known as
a Schrodinger cat state, may be useful for quantum computational processes [11–15]. It has been shown that a
well-controlled entangled atom-photon system is also a
useful tool to investigate quantum decoherence [16]. The
system described in the experiment reported here may
therefore also prove to be a useful tool for systematic
investigation of quantum decoherence with only a handful of atoms.
233001-1
0031-9007=02=89(23)=233001(4)$20.00
PACS numbers: 32.80.Pj, 42.50.Ct
The goal of our study is to observe and subsequently
analyze the propagation characteristics of a weak
Gaussian optical pulse through a high-finesse cavity containing only a few cold atoms N < 10. In order to obtain
an observable signal of such propagation with a reasonable signal-to-noise ratio, the light pulse must interact
with the atoms repeatedly for a sufficiently long period of
time (limited only by the storage time of the cavity). To
achieve this, one must work in the strong-coupling regime
so that relaxation processes due to spontaneous emission
do not cause photons to be scattered by the atom into
vacuum modes and thereby lost from the cavity during
the cavity storage time. Recent progress in highreflectivity mirror technology and atomic manipulation
techniques have provided the means to allow an atom to
coherently interact with the cavity mode in the strongcoupling regime [17,18]. Under these conditions, the detection of the real-time motion of single atoms [19–22],
FIG. 1. The experimental setup. The inset shows the transmittance and refractive index for light propagation through the
cavity. The transmittance splits due to atom-cavity coherent
coupling. The corresponding dispersive property results in
abnormal propagation. According to the slope of the dispersion
curve, subluminal or superluminal propagation takes place at
!1 or !2 .
 2002 The American Physical Society
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PHYSICA L R EVIEW LET T ERS
and the trapping of a single photon have been observed
[23–25]. It is therefore natural to use a strongly coupled
atom-cavity system to investigate optical pulse propagation when the cavity contains a small number of atoms.
To begin, let us consider the case where a few atoms are
located along the optical axis of a microcavity. We denote
the decay rate of atomic polarization by , the vacuum
Rabi frequency by g0 , and the effective intracavity atom
number by N. If g0 > , i.e., the atoms are strongly
coupled to the p
cavity
mode, the vacuum Rabi splitting
is enhanced by N [18], and the dispersion properties of
the atom-cavity system are modified (inset of Fig. 1). A
steep change of dispersion leads to a large variation in the
refractive index, leading to a significant change in the
group velocity vg . At frequency !1 , the index slope is
positive and this results in subluminal propagation, e.g.,
vg < c. On the other hand, at frequency !2 the index
slope is large and negative, resulting in even a negative
group velocity that is commonly referred to as superluminal propagation.
In the experimental setup depicted in Fig. 1, 85 Rb
atoms are first cooled and trapped with a magneto-optical
trap (MOT) comprised of six circularly polarized laser
beams in a cubic arrangement [26]. Typically, 108 atoms
are captured in a few seconds from the background vapor
at a pressure of 109 Torr. We then abruptly extinguish
the two laser beams propagating downward, causing the
atoms to be launched upward into the optical microcavity
located 33 mm above the MOT. The launched atoms are
optically pumped to the F 3, MF 3 state by a laser
before they enter the cavity, and the number of atoms in
the cavity can be well-controlled by adjusting the loading
time of the MOT. We monitor the number of atoms in the
MOT by measuring MOT fluorescence. When the fluorescence signal reaches a preset value, the atoms are
launched upward. This procedure allows us to control
the number of atoms in the cavity from a single atom to
a few tens of atoms.
The optical microcavity (length Lc 70 m) consisting of two concave mirrors has finesse F 2:1 105
and decay constant =2 5 MHz. The cavity resonant
frequency is locked to the F 3 ! F0 4 transition
throughout the experiments using an FM sideband technique [27]. The beam waist w0 for the TEM00 mode of
the cavity is 31 m, and the atom-field coupling constant (vacuum Rabi frequency) g0 =2 is 19.6 MHz for
the F 3, MF 3 ! F0 4, MF 4 transition. Since
the decay rate of atomic polarization =2 is 3 MHz, the
strong-coupling condition is well satisfied.
We derive the probe light from an extended cavity laser
diode whose frequency is precisely controlled using saturated absorption spectroscopy. The spatial mode of probe
laser is purified using a single-mode optical fiber, and the
intensity at the exit of the fiber is actively stabilized using
an acoustic-optic modulator before being sent to the cavity. Before the arrival of the atoms, the probe laser is used
to stabilize the cavity. In this stabilization mode, the
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probe laser intensity is 16 W. It is then decreased to
the pW level, where the average intracavity photon number is 0.4, just before the launched atoms reach the cavity.
The transmitted probe light is monitored using a heterodyne detection technique for which the local oscillator is
generated by frequency shifting the probe laser by
88 MHz. The temporal change in the amplitude of the
88 MHz beat signal (monitored by a spectrum analyzer
with a detection bandwidth of 300 kHz) corresponds to
changes in the transmitted probe intensity.
Figure 2 shows a typical probe transmittance signal
when both the probe and cavity frequencies are locked to
the atomic resonance. Figure 2(a) corresponds to the case
where atoms enter the cavity one at a time. When a single
atom enters the cavity, the probe transmittance decreases
rapidly [19,20] while the atom crosses the cavity mode. In
our case, this transit time is 6 s which is comparable to
the probe pulse width (FWHM 5:6 s). Because the
frequency region in which the steep change in the refractive index occurs is very narrow, ultrashort pulses cannot
be used in the investigation of abnormal light pulse
propagation. We therefore perform the experiments when
the cavity is continuously filled with atoms over a few
milliseconds. With a 3 s MOT loading time, the cavity
mode is filled with several atoms for 2 ms, a time much
longer than the single-atom transit time [Fig. 2(b)].
The intracavity atom number N is determined by measuring the probe transmittance as a function of probe
detuning from atomic resonance. When atoms are present
FIG. 2. (a) The transmitted probe signal when atoms enter
the cavity one after another. (b) The transmitted probe signal
when a group of atoms enters the cavity.
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in the cavity, the transmission of probe light at frequency
!p =2 is given by [18]
2
i! ! a
p
;
T T0 (1)
! ! p
p
where T0 is peak transmittance of the cavity without
atoms, !a =2 is the atomic transition frequency, and
!a g i =2 . When N atoms cooperatively interact with the qsame
light field, the
Rabi fre-
quency is given by g g20 N 2 =4 and is p
well
approximated in the present case ( ) by g g0 N .
Fitting the observed probe transmittance with Eq. (1) can
yield a mean value of the number of atoms (N 6:4 in
the present case).
When several atoms enter the cavity [at t0 in Fig. 2(b)],
we send a Gaussian probe laser pulse at the frequency !p
into the cavity, and measure the arrival time of the probe
pulse after it has traversed the cavity. The delay (advance)
in the arrival time of a pulse propagation through the
cavity with atoms is defined being referred to the propagation time in free space to be t F Lc =vg Lc =c. As
shown in Fig. 3, actually we measured the time delay
(advance) by comparing the arrival time of the light pulse
transmitted through the cavity with atoms to that through
the empty cavity, tobs F Lc =vg tc , where tc is
propagation time through the empty cavity. It should be
FIG. 3. Transmitted probe pulse profiles observed with heterodyne detection method. Solid square: with atoms in the
cavity. Open circle: without atom in the cavity. (a) When the
detuning is =2 53 MHz, 440 ns delay is observed.
(b) When =2 45 MHz, 170 ns advance time is observed.
(c) The delay/advance time as a function of the effective atom
number calculated using a Monte Carlo simulation. The intersections between the curves and the horizontal lines (the
observed delay or advance time) determine the effective number of atoms in the cavity to be N 9:4.
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noted that the empty cavity also induces the delay [8] of
tc Lc
t tobs
c
1 R2
1 R2 4R sin2 Lc
c
Lc Lc
;
c
c
(2)
where
!p !c and R is the reflectivity. R is
calculated
to be 0.99 985 from the relation of
p
F = R=1 R2 .
The pulse shapes depicted in Figs. 3(a) and 3(b) are
the results of accumulation of about 200 times pulse
transmissions, with and without atoms. Because of the
nonlinear nature of atom-light interaction in the strongcoupling regime and the inevitable fluctuations of experiments, the pulse shapes show appreciable deformation. We
fit the observed pulse shapes with Gaussian functions in
order to deduce the delay (advance) times. With this
method we obtained the delay (advance) time. The delay
of tobs of 440 ns at =2 53 MHz [Fig. 3(a)] and
the advance (negative delay time) of 170 ns at =2 45 MHz [Fig. 3(b)] are clearly observed.
We estimate the cavity finesse effect using Eq. (2) for
the present experimental conditions used for Figs. 3(a)
and 3(b) and obtain 0.29 and 0.40 ns, respectively. Since
t tobs t or tobs , t may be well approximated
by tobs .
We have performed Monte Carlo simulations of the
process to compare the observed results with calculations.
We first examine an atom-cavity system with thousands
of atoms randomly distributed p
inside
the cavity mode.
The effective Rabi frequency g0 N can be calculated by
summing the contribution of all atoms [18], giving an expected transmittance
signal TN exp
t tN2 =
p
2
2! , where 2 ln2! is the half-width of the pulse. We
repeated the above procedure 4000 times and then averaged them to obtain the final results of the simulation.
Figure 3(c) shows the delay (advance) time t as a function of the number of atoms. Both the observed delay
(440 ns) and advance ( 170 ns) times give the same
effective atom number of 9.4. (Note the intersections of
the curves and dotted horizontal lines.)
The number of atoms in the cavity (9.4) estimated from
the pulse propagation shows an appreciable disagreement
with that estimated from the Rabi splitting (6.4). This is
due to incomplete optical pumping of atoms into the
MF 3 state. When the population is distributed among
various MF states, the mean Rabi frequency of a group of
atoms becomes smaller than g0 used in the calculation.
For a given Rabi splitting, use of a larger g0 value in the
estimation leads to a smaller atom number.
To further improve the consistencies of the measurements on atom number, we carried out a second set of experiments with the cavity finesse F 7:8 104 (=2 13:8 MHz). In these experiments, the optical pumping
condition was well optimized and the highly sensitive
photon counting detection method was employed. The
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delay of 470 ns at =2 80 MHz, and the advance of
20 ns at =2 96 MHz were observed. The Monte
Carlo simulation on the assumption that all atoms are
pumped into the F 3, MF 3 state gives the intracavity
atom numbers of 15.4 for the subluminal propagation and
16.6 for the superluminal propagation. These values are in
fairly good agreement with the atom number of 15.2
estimated from the Rabi splitting measurement.
The above-described method can be extended to the
case where only one atom is in the cavity. Our calculation
predicts that with properly chosen parameters a single
atom can modify the pulse propagation time, causing up
to 900 ns of delay and 110 ns of advance at the best
condition of our apparatus. The atomic velocity must be
sufficiently slow to perform an experiment with a single
atom, so that the transit time of the atom through the
cavity mode will be much longer than the duration of the
Gaussian probe pulse. We are currently working to improve our experimental setup so that single-atom effects
can be directly observed.
There are other interesting extensions of our experiments such as the creation of an entangled state between
the atom and the transmitted photon (a Schrodinger cat
state). Since a single atom can modify the propagation
time of the optical pulse, if it were prepared in a superposition state, the light pulse would be split into two
propagating parts reflecting the two distinct atomic
states. Such an atom-photon entangled state may be used
in the field of quantum communication and information
technology. It is also possible to investigate the decoherence of an atom-photon entangled state by increasing the
atom number interacting with the cavity in a controlled
way. In the strong-coupling regime, atoms in the microcavity interact cooperatively with the single-photon field,
and the atomic state is well represented by the Dicke state.
However, the phase of the first excited Dicke state may
change dynamically in time because the atoms in the
cavity frequently exchange photons. The more atoms in
the cavity may cause the faster phase evolution of the
Dicke states.
This work was supported by the Core Research for
Evolutional Science and Technology (CREST) of the
Japan Science and Technology Corporation, a Grant-InAid from the Ministry of Education, Science, and Culture, and JSPS Research Fellowships for young scientists.
‡
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*Current address: National Metrology Institute of Japan,
AIST, Tsukuba Central 3, 1-1-1 Umezono, Tsukuba,
Ibaraki 305-8563, Japan.
Electronic address: [email protected]
†
Current address: Toshiba Corporation, 1 Komukai
Toshiba-cho, Saiwai-ku, Kawasaki 212-8582, Japan.
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2 DECEMBER 2002
[26]
[27]
Current address: Tokyo Institute of Technology, 2-12-1
Ookayama, Megro-ku, Tokyo 152-8552, Japan.
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